Vol. 51, No. 1, 1974

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The Garabedian function of an arbitrary compact set

Eric Peter Smith

Vol. 51 (1974), No. 1, 289–300

If the outer boundary of the compact plane set E is the union of finitely many disjoint analytic Jordan curves, the Garabedian function of E is a familiar object. J. Garnett and S. Y. Havinson have each asked whether the Garabedian functions of a decreasing sequence of such sets must converge. The present paper shows that they do converge. This fact leads to a natural definition of the Garabedian function of an arbitrary compact plane set. As an intermediate step, an approximate formula is obtained for the analytic capacity of the union of a compact set E and a small disc not intersecting E.

Mathematical Subject Classification
Primary: 30A40
Received: 26 December 1972
Published: 1 March 1974
Eric Peter Smith